Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,26,25,21)(20,24,28,30)(27,32,31,29)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(2,6)(7,9)(8,13)(14,15)(18,23)(20,28)(21,26)(27,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(4,12)(8,13)(10,16)(14,15)(18,23)(19,25)(20,28)(29,32)(34,35)(39,41)(40,42)(46,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19,22,25)(18,21,23,26)(20,27,28,31)(24,29,30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21)(18,25)(19,23)(20,29)(22,26)(24,31)(27,30)(28,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150338);